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L-functions of Exponential Sums over Finite Fields


Let $\mathbb{F}_{q}$ be a finite field with characteristic $p$. A fundamental problem in number theory is to estimate the reciprocal zeros and poles of L-functions of exponential sums over $\mathbb{F}_{q}$. In this dissertation, we focus on two classical families of exponential sums which have been widely used in the literature. For the type $\mathrm{\RNum{1}}$ family, we compute the weights and $q$-adic slopes of the associated L-functions. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on the weights of toric exponential sums. For the type $\mathrm{\RNum{2}}$ family, the associated L-functions has pure weights. We study the $q$-adic slopes of the reciprocal roots and extend Zhang and Feng's results of Hasse polynomials. Our main tools include Adolphson-Sperber’s work on toric exponential sums and Wan’s decomposition theorems.

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